Design and Reliability Performance Evaluation of Network Coding Schemes for Lossy Wireless Networks

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Design and Reliability Performance Evaluation of

Network Coding Schemes for Lossy Wireless Networks

Li Ma

A thesis submitted in fulfilment of

requirements for the degree of Master of Philosophy

Engineering and Information Technologies University of Sydney

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Abstract

Wireless communication between devices can be lossy in the sense that packet transmissions via a wireless link may fail (so that the packet is lost) due to a number of factors - channel fading, interference or mobility of devices. In some scenarios, the lossy characteristic of wire-less communication can be random, and thus better characterised from a stochastic perspective. This thesis investigates lossy wireless networks, which are wireless communication networks consisting of lossy wireless links, where the packet transmission via a lossy wireless link is successful with a certain value of probability.

In particular, this thesis analyses all-to-all broadcast in lossy wireless networks, where every node has a native packet to transmit to all other nodes in the network. A major challenge of all-to-all broadcast in lossy wireless networks is the reliability, which is defined as the probability that every node in the network successfully obtains a copy of the native packets of all other nodes.

The reliability of all-to-all broadcast in lossy wireless networks can be improved by net-work coding techniques. In this thesis, two novel netnet-work coding schemes are proposed:

1. the neighbour network coding scheme and 2. the random neighbour network coding scheme.

In the two proposed network coding schemes, a node may perform a bit-wise exclusive or (XOR) operation to combine the native packet of itself and the native packet of its neighbour, called the coding neighbour, into an XOR coded packet. By broadcasting the XOR coded packet to other nodes, the reliability of all-to-all broadcast can be improved compared with the corresponding non-coded network wherein a node only broadcasts its native packet. In the first proposed scheme, the coding neighbour is pre-designated; while in the second proposed

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scheme, the coding neighbour is randomly chosen.

The reliability of all-to-all broadcast under both the proposed network coding schemes is investigated analytically using Markov chains and the theoretical analysis is validated by simulations. It is shown that the reliability of all-to-all broadcast can be improved considerably by employing the proposed network coding schemes, compared with non-coded networks with the same link conditions, i.e. same probabilities of successful packet transmission via wireless channels.

The research reveals that the gain in reliability brought by network coding can be signifi-cantly affected by channel conditions, which have however been largely overlooked in existing research in this area. The first proposed coding scheme takes the channel conditions of each node into account in the selection of coding neighbour and further proposes the optimal coding neighbour selection method that maximises the reliability of a given network employing the proposed neighbour network coding scheme.

Moreover, a tuning parameter is introduced in the second proposed coding scheme, which determines the probability that a node performs network coding at each transmission. Further, theoretical solutions to the optimal tuning parameter that maximises the reliability of all-to-all broadcast in networks employing the proposed random neighbour network coding scheme are provided. It is shown that the optimal value of the tuning parameter is also dependent on channel conditions. The observation that channel condition can have a significant impact on the performance of network coding schemes is expected to be applicable to other network coding schemes for lossy wireless networks.

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Acknowledgements

First of all, I would like to express my sincere gratitude to my supervisor, Professor Branka Vucetic. She offers me an opportunity to study at the University of Sydney and has been very supportive throughout my masters candidature. Her great insight into the research area has lead me through this exciting research topic. Her valuable suggestions improve the quality of this thesis. Her attitude of being strict with academic research will shape my career forever.

I am deeply grateful to my co-supervisor, Dr. Zihuai Lin, for his detailed and constructive comments, his invaluable support throughout this work and the financial support. His patience, enthusiasm and immense knowledge helped me all the time of the research and writing of the thesis. He is a supervisor not only for academic research but also for life.

I would like to thank Dr. Zijie Zhang. He has provided detailed discussions in regard to the research topics, and has given invaluable comments on the published papers and the thesis.

I would also like to thank Professor Guoqiang Mao for the kindly discussions and the revisions for the published papers.

Many thanks for editor Margaret Rose Stringer and my friend Nick Morris for proof reading the thesis.

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Statement of originality

I, Li Ma, hereby declare that this thesis, submitted in fulfilment of the requirements for the award of Master of Philosophy, in the School of Electrical and Information Engineering, the University of Sydney, is my own work unless otherwise referenced or acknowledged. The document has not been previously submitted for the award of any other qualification at any educational institution. The results presented in Chapter 4 has been accepted by ICC workshop 2013; and the results presented in Chapter 5 have been accepted by Globecom 2013.

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Related publications

[C1] L. Ma, Z. Lin, Z. Zhang, G. Mao, and B. Vucetic, “Improving reliability in lossy wire-less networks using network coding,” in Proceedings of IEEE ICC workshop CoCoNet, pp. 322-326, 2013.

[C2] L. Ma, Z. Lin, Z. Zhang, G. Mao, and B. Vucetic, “Reliability of all-to-all broadcast with network coding,” in Proceedings of IEEE Globlecom, 2013.

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List of figures

1.1 Illustration of broadcast models. . . 3

1.2 An example of packets exchange between nodes N1 and N2 via N3by (a) tra-ditional store and forward technique, and (b) network coding technique. . . 4

2.1 The classic two-way relaying network applying XOR coding . . . 14

2.2 Illustration of linear network coding, where the column vector assigned to each channel is shown. . . 16

2.3 Butterfly networks where N1and N2transmit packets X1and X2to both N4and N6with traditional store and forward scheme or network coding scheme. . . 18

2.4 An example that network coding used for reduce the number of retransmissions by combining the different lost packets. . . 24

2.5 The comparison of traditional method and the method combining network cod-ing and multipath routcod-ing when a source transmits multiple packets. . . 27

3.1 An example of the all-to-all model where the total number of nodes is n. . . 31

3.2 The relay system for node-to-node transmissions. . . 32

3.3 An example of a packet in a network of four nodes . . . 34

3.4 An illustration of a channel where the packet error rate is qe. . . 35

3.5 The paths between Njand Ni. . . 36

3.6 The system model of all to all communication in one time slot. Note that it is assumed that Nj transmits in this time slot. . . 38

4.1 Illustration of broadcast in lossy wireless network with n nodes applying neigh-bour network coding. . . 41

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4.3 Simulation and theoretical results of the reliability of networks when n= 3, 4, 5. 53 4.4 Network reliabilities under different neighbour selections, where probabilistic

connectivity matrices for n= 4 and 5. . . 55 4.5 The reliability gain of networks employing neighbour network coding scheme

over non-coded network when n= 4 . . . 56 4.6 The reliability gain of networks employing neighbour network coding scheme

over non-coded network when n=4 and 6 . . . 57 4.7 Bounds on the probability that N3receives X1where n= 3 . . . 58 5.1 An illustration of the Markov chain of a network when n= 3. . . 65 5.2 The comparison between theoretical and simulation results of the network

re-liability applying random neighbour network coding when n=3 and 4. . . 73 5.3 The reliability at the end of fourth round when the tuning parameter varies form

zero to one. . . 75 5.4 The reliability at the sixth round when the tuning parameter varies form zero

to one. . . 76 5.5 The expected number of rounds to reach the absorbing state when n= 3. . . 77 5.6 The comparison between reliability of networks with and without network

cod-ing when n= 3 and 4. . . 78 6.1 The performance comparison of two proposed schemes. . . 82 6.2 The performance comparison of two proposed schemes. . . 83 6.3 The performance comparison of two proposed schemes and random linear

net-work coding scheme when the finite filed is GF(2). . . 85 6.4 The performance comparison of two proposed schemes and random linear

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List of tables

4.1 The states of N1for a network with three nodes. . . 43 5.1 The values of the tuning parameter that maximise the reliability at R= 3 to 8. . 76 6.1 The delay for reliability to reach 0.6, 0.7, 0.8 and 0.9 for a network of three

nodes. . . 88 6.2 The delay for reliability to reach 0.6, 0.7, 0.8 and 0.9 for a network of four nodes. 88 6.3 The comparison of the proposed schemes and random linear network coding

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List of symbols

n the total number of nodes in a network

Nk, Nγ, Nθ an arbitrary node

Xk the original packet at Nk

Q probabilistic connectivity matrix

pji the probability that Nireceives a packet broadcast from Njin one time slot

R a round

t a time slot

Nj the transmitter

Ni the receiver

ξk the received packets at Nk

L the total number of states

T the transition matrix at a node

Mµ1

ji, M µ2

ji conditional transition matrices of a node

S the probability vector

ψ the reliability

U the upper bound

L the lower bound

ω the tuning parameter, i.e., the probability to perform network coding µγ,k the index of packet Xγ⊕ Xk

Dj the collection of packets that Njhas successfully decoded

mj the cardinality of Dj

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VA, VB an arbitrary state

vk A the packets of node Nkhas when the network is in state VA

M the transition matrix of a network

Φj VA VB an entry in M

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List of abbreviations

AP Access Point

ARQ Automatic Repeat reQuest

EM electromagnetic

FEC Forward Error Correction

FFNC Finite Field Network Coding

GF Galois Field

LNC Linear Network Coding

LT Luby Transform

MAC Media Access Control

NNC Neighbour Network Coding

RLNC Random Linear Network Coding

RNNC Random Neighbour Network Coding

QoS Quality of Service

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Chapter 1

Introduction

This chapter begins with a brief introduction to wireless communication in Section 1.1, includ-ing the lossy nature of wireless communication and a key mechanism in wireless communica-tion networks - broadcast. Then, network coding techniques, which can be used to improve the performance of wireless communication, are briefly introduced. Then, Section 1.2 introduces some research problems of broadcast in wireless communication networks, followed by a sum-mary of the main contributions of this thesis in Section 1.3. Lastly, the outline of the thesis is provided in Section 1.4.

1.1 Background

1.1.1 Wireless communication

Wireless communication refers to the data communication between devices over wireless chan-nels [1]. Since its invention, wireless communication has created a great impact on everyday life: changing the communication and working habits of a huge number of people. Wire-less communication can provide a large number of applications including Internet access, web browsing, short messaging, file transfer, video teleconferencing, entertainment, sensing, dis-tributed control and health care [2, 3].

One feature of wireless communication that differs from wired communication is the lossy nature, in the sense that packet transmissions via a wireless link may fail and the packet may be lost during the transmission. The lossy characteristic of wireless communication can be caused by a number of issues, detailed thus:

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1.1. Background

• Path loss is the attenuation of signal power when a radio wave propagates through space: it is a major phenomenon affecting wireless connection between a transmitter and a re-ceiver.

• Shadowing is the attenuation of signal power caused by large obstacles between a trans-mitter and a receiver; the variations of the received signal power caused by shadowing are usually modelled by a log-normal distribution.

• Multipath propagation is the phenomenon that a signal reaches the receiver by two or more paths. When a transmitted signal meets an interacting object, reflections and diffractions are created and signals created by reflections and diffractions may also reach the receiver. However, each of these signals may have different amplitudes, phases, de-lays and directions of arrivals; so there is a certain probability that the receiver cannot detect the transmitted signal because the main signal is not strong enough to be detected or because the reflected and diffracted signals may cancel with the main signal.

• Interference refers to the addition of unwanted signals to the useful signal. There are several different types of interference - e.g., co-channel, adjacent channel, inter-symbol, inter-carrier, common mode and conducted interference [4]; they all affect the reception of the signal when the signal-to-interference ratio is small.

In summary, the characteristics of wireless channels cannot, usually, be uniquely deter-mined, and are therefore better characterised from a stochastic perspective. In view of this, research has been conducted on lossy wireless networks, where the end-to-end communication is successful or unsuccessful with a certain probability [5, 6, 7]. This thesis investigates the performance of packet broadcast in lossy wireless networks.

1.1.2 Broadcast

Broadcast is a communication mechanism that disseminates identical data from a source to multiple receivers [1, 8]. There are several basic protocols for broadcast, such as flooding [9, 10] and gossip (epidemic) [11, 12, 13]. Flooding refers to the mechanism that a node sends a message to its neighbours, which in turn forward the message to all their neighbours except the message sender [9]. Using a flooding protocol, a transmitter keeps sending a message to all

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1.1. Background

its neighbours in the assigned time slots. Yet another broadcast protocol is gossip (epidemic). In this protocol, each node contacts only one or a few nodes in each round, and exchanges information with these nodes. Gossip protocol has been widely applied in ad hoc networks [13, 14].

There are two main types of broadcast models, which are the one-to-all model and the all-to-all model. The one-all-to-all model represents the mechanism of transmitting information from one source to all other nodes in the network, as illustrated in Figure 1.1 (a). It is a model widely considered and referred to in the open literature. In an all-to-all model, every node is a source and transmits information to all other nodes in the network, as illustrated in Figure 1.1 (b).

Figure 1.1: Illustration of (a) a one-to-all model where a single node broadcasts to all other nodes; (b) an all-to-all model where every node broadcasts information to all other nodes.

Traditionally, the all-to-all broadcast is applied to routing algorithms, such as finding routes in adaptive routing mechanisms [15] and in ad-hoc routing mechanisms [16]. It is also applied to content distribution [17], where content distribution refers to the delivery of information to a large number of users in a network. In addition, it can be used for communication among users to update the status of all users [18]. More recently, it has become regarded as a key mechanism for data communications in intermittently connected ad hoc networks, e.g. an airborne network [19] or a delay tolerant network [20].

In this thesis, we investigate all-to-all broadcast in lossy wireless networks, where network coding is applied at each node to improve the reliability performance.

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1.1. Background

1.1.3 Network coding

Network coding [21] is a technique allowing nodes in a network to intelligently combine and compute independent incoming information flows into an outgoing information flow. In com-parison, in traditional networks without network coding, nodes can only forward information without changing the content of the information using the store and forward technique.

In illustration of this, Figure 1.2 shows the packet exchange between nodes N1 and N2: node N1has a native packet X1to send to N2, while N2has a native packet X2to send to N1.

Figure 1.2: An example of packets exchange between nodes N1and N2via N3by (a) traditional store and forward technique, and (b) network coding technique.

In Figure 1.2 (a), the network applies the traditional store and forward scheme: the relay node N3 simply forwards packets X1 and X2 separately in different transmissions. The total number of time slots required for N1 and N2 to exchange their packets is four, and the total number of transmissions required is four. In the network, where exclusive or (XOR) network coding is applied, as shown in Figure 1.2 (b), the intermediate node N3is allowed to combine packets X1 and X2 to generate an encoded packet, e.g. X1⊕ X2. Then, this coded packet is transmitted. Lastly, upon receiving the encoded packet, both nodes N1 and N2 can decode the required packet using their native packets. The required time slots and total number of transmissions for the exchange of packets are both three.

This demonstrates that the network coding technique can reduce the required number of time slots and total number of transmissions compared to the traditional store and forward technique; thus network coding can be seen to assist in the sharing of available network re-sources - e.g., bandwidth and energy [22]. A review of network coding techniques and the

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1.2. Research problems

benefits that network coding techniques bring to a network will be provided in Chapter 2 .

1.2 Research problems

1.2.1 Reliability

There are a number of performance metrics for a communication network, such as delay, band-width efficiency, energy efficiency and reliability. Due to the lossy nature of wireless communi-cation, the packet transmitted from a source node may not always be able to reach its intended destination(s). Therefore, reliability of information transmission is a key research problem in lossy wireless networks. In this thesis, the reliability is defined as the probability that a piece of information will reach its intended destination(s).

Numerous error control methods have been developed to attempt to ensure a high relia-bility communication in lossy networks. There are several error control approaches; the basic ideas of these methods are to intelligently retransmit a piece of information or to perform error correction. One common method is the Automatic Repeat reQuest (ARQ) [23], which enables receivers to send the receiving status (received/not received) back to the transmitter; then the transmitter decides whether or not to retransmit the same information, based on the feedback of the receiving status. However, this simple repetitive retransmission scheme results in delay and is a non-efficient use of bandwidth in the processes of sending the receiving status and retransmissions.

It has been shown that coding techniques can improve reliability [5], bandwidth efficiency [24] and reduce delay [25]. The challenge lies in finding suitable coding methods to improve reliability while taking into account delay and throughput.

There has been some research focusing on the expected number of retransmissions for in-formation to reach its intended destinations in the one-to-all model [5]. However, the reliability in the all-to-all model is overlooked in the literature. In addition, the reliability after each re-transmission is less understood; and it is useful to some real world applications - e.g., delay constrained transmissions.

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1.2. Research problems

1.2.2 Coding design

In wireless communication networks, coding operations are designed for data compression, cryptography and reliable transmission [26].

Source coding refers to the process of coding at the source of the data before the data is stored or transmitted. It involves encoding information using fewer bits than the original representation; therefore, it is popularly referred to as data compression [27].

Channel codingis the scheme that transmits redundancy information so that the noisy chan-nel can behave like a noiseless chanchan-nel [28]. The fundamental chanchan-nel codes are block codes and convolutional codes. In particular, Turbo codes [29], which are a successful class of con-volutional codes, introduce a way of designing codes with feasible decoding complexity.

Fountain codes have been studied extensively in the literature for their channel coding applications. Fountain codes are efficient and robust solutions for reliable transmission over erasure channels through which a transmitted packet can be either received without error or not received. [30]. Metaphorically speaking, the encoder is the source of a fountain providing an endless supply of water drops (encoded packets), and these encoded packets are a limitless and rateless sequence of encoded symbols from a set of symbols transmitted at the source. Then the receiver needs to collect a certain number of encoded packets to decode the whole set of original source symbols.

Among all fountain codes, the first practical and successful class are the Luby Transform (LT) codes [31]. Using LT codes, the source node generates an encoded symbol from a set of input packets as follows:

• The source node randomly chooses an integer d, which is called the degree of the encod-ing packet, accordencod-ing to a degree distribution ρ(d). The optimal degree distributions in various systems have been investigated extensively in the literature [32].

• The source node chooses uniformly at random d distinct input packets; and encode these randomly chosen packets by performing an exclusive or (XOR) operation on them. Fountain codes are near optimal for every erasure channel in the sense that they can be applied to an erasure channel regardless of the statistic of the erasure event.

However, fountain codes are not sufficient in distributed coding, and they are not always sufficient in the all-to-all model. This is because in fountain code, it is assumed that the

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com-1.3. Main contributions

plete set of packets to be broadcast is available in each transmission. However, the packets to be transmitted at a source node in an all-to-all model vary over time: they include not only the packet initially owned by the source node, but also the packets received and decoded. There-fore, in an all-to-all model, there may not be sufficient numbers of packets at a node to achieve a specific coding degree d required by fountain codes.

Network codingtechnique enables an intermediate node to combine and process received information. It is suitable for an all-to-all model where the encoding of the packets to be broadcast at a node involves the packets previously received by the node. However, network coding schemes are mostly designed and applied to one-to-one and one-to-all models in the open literature, as will be introduced in Chapter 2.

This thesis focuses on designing network coding scheme suitable for the all-to-all model, where the encoded packet is determined adaptively at a source depending on its received pack-ets without the aid of feedback information. Moreover, these schemes should adapt to lossy networks with different channel conditions.

1.3 Main contributions

This thesis proposes two novel XOR based network coding schemes for all-to-all broadcast in lossy wireless networks, called the neighbour network coding scheme and the random neigh-bour network coding scheme, which will be introduced in detail in Chapters 4 and 5 respec-tively.

Both schemes are adapted to the time-varying status of the packets received at each node. More specifically, in both schemes network coding is performed at a source node between the native packets of the source node and another selected node, namely the coding neighbour, where the native packet of a node is the non-coded packet that the node initially has. It is worth mentioning that the selection of the coding neighbour does not rely on feedback information of the receiving status (received/not received), but it is affected by the end-to-end connection probabilities among nodes. The end-to-end connection probabilities of pair of nodes reflects the channel conditions, and the means to obtaining these probabilities will be introduced in Chapter 3.

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1.3. Main contributions

small. Using the proposed coding algorithms, the successful decoding of a coded packet only requires that one of the two native packets forming the coded packet has already been success-fully received or decoded. For example, Xzcan be decoded from packets Xz⊕ Xk and Xk by performing (Xz⊕ Xk) ⊕ Xk, where z , k.

In the neighbour network coding scheme, each node selects another node as its coding neighbour. As soon as the native packet of the predetermined coding neighbour is received or decoded, the node begins to broadcast an encoded packet which is the bitwise XOR between the native packets of itself and that of its coding neighbour. On the other hand, in the random neighbour network coding scheme, each node randomly chooses 1) whether or not to perform coding according to a tuning parameter, which is described in detail in Chapter 5; and 2) with which packet to perform coding on-the-fly according to the packets that it has received and decoded. Therefore, no encoding delay is introduced in both schemes.

The reliability of all-to-all broadcast models applying the proposed schemes is investigated, where the reliability of all-to-all broadcast is defined as the probability that every node in the network receives or decodes the native packets of all other nodes. It is shown that the network reliability can be improved considerably by both proposed network coding schemes.

Further, optimisations are carried out to maximise the reliability in networks applying the proposed network coding schemes. For the first scheme, optimal neighbour selection rules are proposed; while for the second scheme, the optimal value of the tuning parameter is derived.

Lastly, the reliability performance of a network applying the proposed schemes are com-pared with each other and further comcom-pared with the performance of the random linear network coding scheme. It is shown that the proposed network coding schemes, when the coding neigh-bour selection or tuning parameter is optimised, can outperform the popular random linear net-work coding scheme where the randomly generated coefficients are chosen from the finite field GF(2).

1.3.1 Summary of the contributions

We proposed two novel network coding schemes which take advantage of the simplicity of the XOR-based coding. In the neighbour network coding scheme, each node selects another node as coding neighbour and will broadcast a coded packet as long as the native packet of the coding neighbour is received or decoded. This coded packet is the bitwise XOR between

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1.4. Thesis outline

the native packet of the source node and the native packet of its coding neighbour. In the random neighbour network coding scheme, each node performs coding according to a tuning parameter, where the tuning parameter is the probability that all nodes perform network coding. Network coding is performed between the native packet of a source node and another native packet which is selected on-the-fly according to the received packets. It is observed that the key in the design of network coding schemes for lossy wireless networks is to take channel conditions into consideration. Both of the proposed schemes can be adjusted by selecting the optimal coding neighbours in the first scheme or the optimal value of the tuning parameter in the second scheme to maximise reliability in networks with different number of nodes and different channel conditions. The encoding and decoding processes of the proposed schemes are of low computational complexity. It is shown by theoretical analysis and simulations that the reliability of networks applying the proposed schemes is considerably improved compared with that in the networks without network coding and that in the networks employing random liner network coding with GF(2). For example, the proposed schemes are able to achieve reliability improvements of 272.34 percent over the non-coded networks, and 100.90 percent over the random linear network coding with GF(2). It is worth noting that the theoretical framework and analysis methods established in this thesis can be easily modified and extended to examine the reliability of networks applying other XOR-based network coding schemes

1.4 Thesis outline

The thesis proposes two novel network coding schemes, and each will be introduced in one individual chapter. The rest of the thesis is organised as follows:

• Chapter 2 reviews the fundamental network coding schemes, and evaluates their perfor-mance when applied in different network topologies under various traffic configurations. Then, the benefits made by network coding are summarised. Lastly, the challenges in code design are summarised.

• Chapter 3 introduces the system model. Firstly, it gives a description of the communica-tion strategy in an all-to-all model. Secondly, the structure of a node is presented and the functions of some main parts are described. Thirdly, the metric used to measure channel

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1.4. Thesis outline

conditions from one node to another is introduced. Lastly, the data flow diagram of the system in one time slot is presented.

• Chapter 4 presents a novel neighbour network coding scheme. The network reliability under this coding scheme is evaluated analytically and the theoretical analysis is vali-dated by simulations. It is shown that network reliability can be improved considerably by the proposed neighbour coding scheme compared to the reliability of the correspond-ing non-coded network. Further, closed-form upper and lower bounds on the network reliability are derived. Moreover, the optimal neighbour coding selection rules that max-imise the reliability of a given network are introduced.

• Chapter 5 presents a novel random neighbour network coding scheme. The network reliability is analysed theoretically, the optimal value of the tuning parameter that max-imises the reliability is derived and the theoretical analysis is validated by simulations. Furthermore, it is shown that the random neighbour network coding scheme can improve the network reliability significantly.

• Chapter 6 compares the proposed schemes with each other in regard to their similarities, differences and reliability performance. Further, both schemes are compared with ran-dom linear network coding scheme in relation to their reliability and delay performance. • Chapter 7 concludes the thesis and proposes possible future work.

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Chapter 2

Network Coding

Traditionally, a network delivers information using the store and forward technique. On the other hand, network coding technique breaks this tradition and allows nodes in a network not only to store and forward but also to intelligently combine and compute the independent in-coming information flows into an outgoing flow [21].

A lot of research has been carried out on network coding theory. Among them, there are several successful categorises of network coding schemes. Section 2.2 reviews some basic net-work coding schemes, including XOR based netnet-work coding, linear netnet-work coding, random linear network coding and distributed random linear network coding.

Along with the development of network coding theory, the problem addressed by network coding is extended from the original multicast capacity to energy efficiency, delay, reliability, etc. Section 2.3 reviews the benefits brought by network coding techniques.

Further, the challenges in the design of network coding schemes are summarised in Section 2.4.

2.1 Background

Terminology. The topology of a communication network can be represented by a graph. A graphconsists of vertices or nodes, and lines connecting the nodes are called edges, channels or links. A graph may be undirected, meaning that the links pointing from either one node to the other node within the same pair of nodes have no distinction. On the other hand, in a directed graph, the edge has a direction associated with it pointing from one node to another.

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2.1. Background

The concept of network coding is first proposed in 2000 [21] to solve the problem of defin-ing admissible coddefin-ing rate region, where the admissible coddefin-ing rate region is the region rep-resented by the points in space that can achieve an arbitrarily small decoding error probability by block codes. It reveals that it is in general not optimal to multicast information by store-and-forward. Rather, by employing network coding where the information can be coded and combined at a node, bandwidth efficiency can be improved. In [21], networks are represented by directed graphs, where the edges are error-free. This model can be used to represent a wired network where links are lossless, e.g., an Internet backbone.

Along the same direction, the network coding theory in error-free directed graphs is in-vestigated. In [33], the linear network coding scheme and in [34], the random linear network coding scheme are proposed, showing that improvements in capacity can be made by network coding. These schemes lay the foundation of network coding theory and they will be reviewed in detail in Section 2.2.

On the other hand, the network coding technique is extended to and examined in other network scenarios. For example, it is applied to networks with transmission errors and to undirected networks.

When a packet is transmitted in lossy networks, errors may occur due to channel fading, interference, or mobility of devices. One possible solution for reliable transmission is to de-velop techniques that counteract the errors. The pioneering work employing network coding to solve this problem is the network error correction code which is proposed in [35] and later improved in [36, 37]. When information is transmitted over an individual link experiencing errors, network codes are applied for error correction. A network code is defined as t-error correcting if it can correct every one of a total of τ errors occurring in the network during transmission, where τ ≤ t, i.e., the total number of errors in the network is less than or equal to the maximum number of errors that can be corrected by the code, which is t. It is shown that the network error correction is a generalisation of point-to-point error correction and it has become an instrument for the effective use of network coding in lossy networks [38]. Following this idea, a lot of research has been conducted to examine the network error correction code of its performance bounds [39], code constructions [40] and decoding algorithms [41]. For example, the separating principle of channel coding and network coding is defined in [42]. By doing so, the information can be transmitted through a lossy link in a lossless way.

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2.1. Background

Note that the aforementioned research examines network coding in directed networks. The network coding technique is first applied to undirected networks in [43]. It is shown that net-work coding can also bring benefits in terms of throughput in undirected netnet-works. However, unlike directed networks where the gain in throughput brought by network coding is not finitely bounded [44], the gain for undirected networks applying network coding over non-coded net-work is upper bounded by a constant factor of 2 [43]. Following this net-work, the bounds on the throughput are further analysed in [45, 46, 47]. In [47], it is shown that the upper bound derived in [43] is generally true in undirected network topologies with any link capacity configuration, any multicast group size, and any source information rate.

Despite the fact that network coding technique is proposed to be applied at network layer initially, its vast benefits draw attention of researchers to apply it at other protocol layers. Inspired by the combination of transmitted packets from different senders, in physical layer, the electromagnetic (EM) waves transmitted by different senders may be added at a relay node. Therefore, instead of processing one EM wave while treating other EM waves as interferences, multiple EM waves may be combined for computing an output signal [48]. This network coding scheme is referred to as physical layer network coding scheme and its publication opens a new research area. The performance of physical layer network coding scheme is examined extensively in terms of throughput [48], reliability [49] and delay [50]. For example, it is applied to a two-way relaying network in [51]. By examining and comparing the sum rate and sum bit error rate of different transmission schemes, the benefits of physical layer network coding are demonstrated.

Other problems relating to network coding techniques are widely examined in the open literature in regard to the memory size [44], number of nodes to perform encoding [52] and computation complexity [53] etc. Along with the development of network coding theory, its application is extended and shifted from the multicast throughput originally to other perfor-mance and other traffic configurations. For example, network coding is applied to storage networks, which are normally modelled by combination networks. The combination network Cn,k refers to a network with one source, n relays and

_k n

receivers. The capacity gain brought by network coding in these networks are examined in [54]. Further, a deterministic approach to achieve the multicast capacity limit in these networks is presented in [55].

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2.2. Network coding schemes

2.2 Network coding schemes

This thesis considers network layer network coding, which concerns the data transmission in the packet level. There are several classes of network layer network coding schemes, which are explained in the following.

2.2.1 XOR based network coding

In the XOR based network coding schemes, bitwise XOR are performed among all or a certain set of packets at a node. Consider the example where nodes N1 and N2exchange packets X1 and X2via a relay N3, as shown in Figure 2.1. Assume that node N1has the native packet X1 while node N2has the native packet X2initially.

Both nodes transmit their native packets to the relay node individually in the first stage. At the relay node, an XOR coding is conducted between the received packets X1 and X2. Then, instead of transmitting two packets separately, N3 transmits the coded packet X1 ⊕ X2 in one transmission. Finally, at a destination node, the coded packet can be decoded with the assistance of its native packet. For example, at N1, the required packet X2 can be decoded by performing (X1⊕ X2) ⊕ X1.

Figure 2.1: The classic two-way relaying network applying XOR coding

XOR coding has drawn an increasing attention owing to its simplicity in both encoding and decoding processes. For example, COPE [56], which is the practical network coding scheme for wireless mesh networks, applies the XOR coding. In COPE, each encoded packet can be decoded upon arriving at a node. In order to achieve this, a router encodes packets with the assistance of MAC layer feedback information. By applying COPE, the throughput of the network can be improved.

This thesis focuses on the reliability benefit made by XOR based network coding when utilised in broadcasting without feedback information.

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2.2.2 Linear network coding

Linear network coding has been proved to be able to achieve capacity limit from the source to each receiving node in multicast networks [33]. The capacity limit, given by max-flow min-cut bound [57], is a conclusion from graph theory. More specifically, the information from a source to a destination can be delivered through the network at a maximum rate equal to the min-cut between them [21].

The data transmission using linear network coding scheme is specified in a multicast model, where a source multicasts to multiple receivers. As shown in Figure 2.2, the source N1transmits information to destination nodes N6and N7.

Each link is assigned with a column vector of d dimensions, where d is the maximum value of the max-flow of every non-source node. The entries of these vectors are selected from a finite filed, say Galois field GF(q), where q is an arbitrary positive integer. In the example shown in Figure 2.2, d = 2. In addition, the selected finite field is GF(2). N1N2 is assigned with vector [1 0]T, where N1N2stands for the channel from N1to N2.

Additionally, the vector assigned to an outgoing link from a node is the linear combination of the corresponding vectors assigned to the incoming links. As shown in Figure 2.2, the vector assigned for N4N5is [1 1]T which is equal to [1 0]T+ [0 1]T, where [1 0]T and [0 1]T are the vectors assigned for two incoming links of N4, which are N2N4and N3N4respectively.

Further, the information to be transmitted is encoded into a d-dimensional row vector. The data flow on a channel is represented as the matrix product of the information row vector with the assigned column vector of the channel. In this case, the data sent on an outgoing channel of a node is a linear combination of the data sent on the incoming channels of the node. In Figure 2.2, suppose the data transmitted by N1 is [b1 b2]. Then, the data sent on channel N1N2 is represented by [b1 b2] × [1 0]T, which is b1. Similarly, at N4, the data transmitted is represented by [b1b2] × [1 1]T, which is b1+ b2.

Lastly, at a destination node, the information of the source can be retrieved from the re-ceived packets with the aid of the channel vectors [33].

Following [33], [58] solves an important problem of finding the encoding functions at intermediate nodes, where the encoding functions are used to determine the vectors assigned to each channel. The basic idea in [58] is to match the network coding solution to a solution of

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Figure 2.2: Illustration of linear network coding, where the column vector assigned to each channel is shown.

a set of linear equations, because every solvable multicast network has a scalar linear solution over a sufficiently large finite-field alphabet. However, linear network coding is not always sufficient [59], because some networks do not have linear solutions in any field.

2.2.3 Random linear network coding

In order to apply linear network coding to networks with unknown or changing topologies, random linear network coding is proposed [34]. Unlike linear network coding where every link is assigned with a planned and deterministic vector, random linear network coding enables nodes to randomly select a coefficient for each incoming packet over a finite field.

Consider the same network topology as that in Figure 2.2. When random linear network coding is applied, the message transmitting on channel N1N2can be represented by ξ1b1+ξ2b2, where ξ1, ξ2 ∈ GF(q), and q is an arbitrary positive integer. In fact, the message transmitting on every channel can be represented by the same expression. The reason is that in linear algebra, the result of arithmetic operations of two values from a finite filed falls into the same filed. For example, the transmitting message on channel N4N5is ξ5(ξ1b1+ ξ2b2)+ ξ6(ξ3b1+ ξ4b2), which can be represented by ξ7b1 + ξ8b2, where ξ1, ξ2, ξ3, ξ4, ξ5, ξ6, ξ7, ξ8 ∈ GF(q). Therefore, in

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2.3. Benefits made by network coding

general, when a source transmits X1, X2,. . . , XM and random linear network coding is applied, the data transmitting on a channel can be represented byPM

i ξiXi, where ξi∈ GF(q).

The encoding functions which determine the coefficients of the channels are required to be delivered to the destination nodes to allow them retrieve the coded packets. Then, at a receiver, decoding can be performed by Gaussian elimination when enough number of linear independent coded packets are received. It is worth noting that the probability to achieve the theoretical multicast capacity is exponentially approaching one with regard to the field size [34].

2.2.4 Distributed random linear network coding

In both [33] and [34], the global knowledge of coding coefficients is required for decoding. However, it can be difficult to achieve in reality. To solve this problem, a distributed network coding scheme is proposed [60]. In [60], a data-aided transmission scheme is applied, where each outgoing data packet flowing on an edge of the network includes a packet header contain-ing the coefficients of the linear combination of the coded packets. With the cost of overhead, network coding can be utilised in a decentralized manner in networks. Moreover, a generation tagis introduced in [60] to mark the coded packets that are related to the same set of source information. This method solves the problem of synchronisation of incoming and outgoing packets belonging to the same set of packets.

However, these methods not only increase the time complexity of network coding-based approaches, but also increase the complexity to implement them practically.

2.3 Benefits made by network coding

Bandwidth and energy supply of nodes are primary resource constraints in wireless communi-cation [61]. Network coding, by allowing different information flows sharing these resources, can improve bandwidth and energy efficiencies [22]. Besides, network coding has show benefit in reducing the delay of receiving a set of packets. Moreover, it has been utilised as an error control method for reliable transmission.

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2.3.1 Bandwidth efficiency

The first paradigm where network coding shows its advantage is the improvement in capacity when it is applied to error-free multicast [21]. The coding at the intermediate node enables mul-tiple information streams to be transmitted simultaneously, resulting in a substantial bandwidth improvement over networks applying transitional store and forward scheme. Since its publi-cation, the bandwidth benefit made by network coding is examined extensively and network coding has been included in many bandwidth-efficient transmission schemes.

The basic idea

The following illustrates the idea behind the improvement in bandwidth efficiency brought by network coding. Figure 2.3 shows the packet transmissions on butterfly network where sources N1 and N2 transmit native packets X1and X2receptively to both destinations N4and N6. The network coding scheme utilised here is XOR coding. In addition, the store and forward scheme is drawn for comparison.

Figure 2.3: Butterfly networks where N1and N2transmit packets X1and X2to both N4and N6 with traditional store and forward scheme or network coding scheme.

In Figure 2.3 (a), the network applies the traditional store and forward scheme. The central node N3 can only forward one packet at one time. Therefore, it takes two transmissions for N3 to broadcast both packets X1 and X2. With the use of network coding, as shown in Figure 2.3 (b), intermediate node N3is allowed to combine packets X1and X2then to broadcast this network coded packet in one transmission. Finally, the coded packet can be decoded at nodes

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N4 and N6. It can be seen that the number of transmissions for the destinations to have both packets is reduced. Therefore, the bandwidth efficiency is improved by network coding.

Following this idea, network coding has been applied in ad hoc networks [62, 63, 64, 65, 66, 67] and wireless mesh networks [68] to increase bandwidth efficiency by reducing the number of transmissions.

In [67], network coding is applied to one-to-all broadcast in multi-hop networks where a node predetermines and selects the neighbouring nodes to forward the broadcast packet. Net-work coding is performed according to the proposed algorithms which rely on two-hop topol-ogy information. It shows in networks applying the XOR based network coding algorithms that the number of transmissions can be reduced up to 45 percent compared to the non-coding approach. In [63], two-way relaying network, as shown in Figure 2.1, is examined. The in-formation exchange rate is calculated when network coding is applied. It is shown that the exchange rate is improved compared to conventional solutions that separate the processing of two unicast sessions.

The average throughput has been analysed in a multicast model where there are n receivers and either linear network coding scheme or random linear network coding scheme is applied [69]. The average throughput is the averaged rate that individual receiver experiences. Using linear programming formulations, it is shown that network coding offers benefit in average throughput proportional to √n. Furthermore, when a network has a large number of nodes, the average throughput in the network with network coding at most doubles compared to that without network coding.

Other traffic configurations

The gain in bandwidth efficiency made by network coding is not constrained to the case of mul-ticast [69], but also applicable to other traffic configurations, such as multiple unicast sessions [70].

COPE [56], as introduced in Section 2.1, is a technique that enhances the throughput of wireless multi-hop networks when the traffic is transmitting in unicast fashion. Based on that, a theoretical analysis of the throughput gain in multi-hop wireless networks which apply the COPE-type opportunistic network coding, is presented in [70]. Further, coding-aware rout-ing is proposed in [70] and compared with interference-aware routrout-ing, where codrout-ing-aware

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routing refers to finding the path that increases coding opportunity. Finally, an optimisation is conducted by linear programming to find the maximum unicast throughput. It is shown that a route selection strategy that is aware of network coding opportunities leads to higher end-to-end throughput when compared to coding-oblivious routing strategies.

The throughput of the network coded unicast sessions can be further improved by routing [71, 72, 7]. In [71, 72], random linear network coding is augmented upon existing routing protocols. It allows the source node to continuously send the random linear coded packets through multiple opportunistic paths until the destination collects a sufficient number of packets for decoding. This protocol is particularly suitable for delivering files of medium to large size [71, 72]. In [7], a protocol is proposed to optimise the multipath routing and code rate to maximise throughput when network coding is applied to lossy wireless networks.

Undirected networks

The throughput benefits are examined in undirected networks. In [73], the upper bound of throughput gain in a special network topology, i.e. the combination network topology, is stud-ied. Linear network coding is applied to combination networks. By studying the cost of min-imum multicast tree, it is shown that network coding can improve throughput of combination networks upper-bounded by a factor of 9/8.

2.3.2 Energy efficiency

Networks with network coding has shown advantage in energy efficiency [64, 74, 75, 17, 76], where energy efficiency is usually measured by the energy used to transmit one bit of informa-tion.

Multicast

In a conventional network where a node only store and forward incoming information, the energy efficient routing relies on finding the minimum-energy multicast tree [77]. There is some research focusing on computing the minimum-energy multicast tree [78, 79], which is, however, usually NP-hard.

An alternative method for minimum-energy multicast in a mobile ad hoc network is pro-posed in [64]. It considers a layered model of wireless networks, and then accordingly

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con-2.3. Benefits made by network coding

structs a set of realisable graphs, where each edge of the graph is assigned with the energy-per-bit for the corresponding channel. Then, the bit-rate demand on the edges is characterised by the edge-wise maximum of flow from the source to each destination. It is shown that the minimum energy multicast energy-per-bit can be obtained using linear programming, which can be attained by performing network coding, but not routing.

Other traffic configurations

The energy efficiency of a two-way relaying system is examined in [74], where a network coding scheme namely a hybrid automatic repeat request with incremental redundancy scheme is applied. It calculates the average energy consumption for signal transmission over block fading channel and concludes that network coding brings benefit in terms of energy efficiency to the two-way relaying networks.

The energy efficiency of broadcasting in an all-to-all model is examined in [75, 17, 76]. The network considered in [75] has a circular topology while Ref. [17] examines circular, rectangular grid and random network topologies. In [75], both linear network coding and random linear network coding schemes are applied to ad-hoc networks. The simulation results show that there are significant improvements in energy efficiency for these coded networks over networks applying the traditional store and forward approach. In [17], algorithms are proposed to improve energy efficiency by network coding. These algorithms aim at minimising the number of transmissions. Because it is assumed that each transmission consumes the same amount of energy, the total energy is proportional to the number of transmissions. In addition, the energy efficiency under these algorithms are evaluated, showing that network coding can improve energy efficiency by a constant factor in fixed networks, and by a factor of log n in dynamically changed networks, where n is the number of nodes in the network [17].

2.3.3 Delay performance

Network coding technique does not always bring benefit in the delay point of view, because delay may suffer from the encoding and decoding procedures which need to collect enough packets to proceed. There is a trade-off between throughput efficiency and decoding delay: the larger throughput a system has, the more packets need to be coded together, which in turn

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packets in order to decode a single packet [80]. In delay sensitive applications, the delay performance should be taken into consideration.

The average delay

Network coding can be used to reduce the delay in scenarios of large files transmissions. The system considered in [25] is a single-hop one-to-all model, where the source transmits files to multiple receives via time-varying lossy channels. Closed-form expression for the delay per-formance is given. It is shown that network coding can bring arbitrary gain in delay by scaling the system parameters. After comparing different network coding schemes, it is concluded that random linear network coding has the minimum mean completion time, where the completion time is the time that all receivers receive the complete file. However, these results are not feasible in the case of packet streaming.

In lossy networks, the mean time to complete the transmission of a block of packets from a source to all receivers is examined in [81]. Random linear network coding is used to encode the block of data, and the receiver can recover the information when enough coded packets have been collected. Therefore, it only needs to record the required number of packets to perform decoding rather than keeping track of which packets have been received. This required number of packets is included in the feedback information that is sent occasionally. A Markov chain is used to describe the transmission process where the state represents the number of packets required for decoding at each receiver. Then, the all-to-all model applying random linear network coding is analysed in [82]. The block of data to be encoded is ready in the first transmission. In comparison, the encoding is conducted on dynamically received packets in schemes proposed in this thesis.

In [83], the relation between the average delay and some factors, such as scheduling and the size of coding buffer is examined. The system model is that one transmitter transmits data stream to a set of one-hop receivers. In order to minimise the average delay per received packet at all receivers, an adaptive method is developed to find the maximum number of packets that can be encoded using random linear network coding. Moreover, a network coding scheme is proposed aiming at further reducing the delay by sending non-coded packets. The upper and lower bounds of the delay performance in multicast downlink transmission are derived [84]. The expected delay is also considered. The expected delay is the average time to receive

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a block of packets that enables decoding. Additionally, the expected encoding delay and the expected transmission delay at the transmitting node are examined. It is shown that the network coding scheme proposed in [83] enables a network to achieve the theoretical lower bound of the expected delay.

Delay distribution

The design of some real-time applications requires not only the average decoding delay but also the worst-case delay, which can be inferred from the complete probability distribution of delay. It is worth to note that in [85], delay probability distribution is investigated. Delay distributionis the probability of successful decoding of all packets at individual delay, which is similar to the reliability after each transmission considered in this thesis. Ref. [85] considers a one-to-all model where a single source broadcasts packets encoded by random linear network coding scheme to all other nodes over erasure channels. The broadcast message is available prior to the first broadcast. Markov chain is used to analyse a network with three nodes only and a brute-force method is proposed for four nodes. In contrast, in the model considered in this thesis, the message to be broadcast by a source node at a given time slot depends on the packets that the source node has received in previous time slots. Consequently, the message to be broadcast by a source node is varying over time, which is different from previous work [85] where the message broadcast by a source node does not vary over time. Moreover, this thesis considers all-to-all broadcast in networks with arbitrary number of nodes.

The exact probability to decode N linearly independent packets among K received packet at a receiver is examined in [86]. It is assumed that the transmitted packets are encoded by random linear network coding and the coefficients can be selected from a finite filed of arbitrary size. The number of received packets is proportional to the delay. Therefore, the methods introduced in this paper can be used to measure the delay distribution.

2.3.4 Reliability

Another important benefit that network coding technique is able to bring to a network is to im-prove the reliability. Conventionally, in a non-coded wireless network, if a transmission fails (lost or dropped due to unsuccessful error correction), then successful reception of a packet

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search has been conducted to reduce the number of retransmissions while maintaining network reliability. Most recently, network coding has been utilised as an error control method for loss recovery in lossy networks.

Combination of retransmission and network coding

Network coding is combined with retransmission techniques to improve the reliability of pack-ets transmissions to multiple receivers. The XOR based network coding is applied to lossy networks to reduce the number of retransmissions.

When packets are broadcast in lossy wireless networks, many destinations may have dis-joint lost packets. Instead of transmitting individual lost packet separately, the sender may encode the lost packets of different receivers and broadcast the coded packet to all receivers. Then at an individual receiver, the lost packets can be recovered with the knowledge of previ-ously received packets. In this case, multiple receivers may recover their lost packets in one transmission and therefore, the total number of retransmissions can be reduced. In illustration of this, an example is given in Figure 2.4, where node N1 transmits packets x, y and z to N3, N4 and N5 via a relay N2. After broadcasting, each node has different lost packets, which are y, z and z for N3, N4and N5respectively. If network coding is applied, the relay node N2can retransmit a coded packet y ⊕ z, allowing every destination node to recovery their lost packets in one retransmission.

Figure 2.4: An example that network coding used for reducing the number of retransmissions by combining the different lost packets of different receivers.

The reliability of networks applying network coding are examined in [87, 24, 88] where a reception report mechanism is employed to inform the sender of lost packets of each receiver.

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In this case, the encoded packets can be generated using the knowledge of lost packets at each receiver.

The network coding aided ARQ is applied to access point (AP) based networks in [87]. More specifically, a source node employs network coding to broadcast a selected combination of unsuccessfully received packets of different receivers. In [87], all users listen to all the packets, and intended users may decode the network-coded packet using the overheard packets. In this way, the number of retransmissions can be reduced.

Packet retransmission algorithms based on network coding for wireless broadcast in one-to-all model are proposed in [24]. An XOR coding is employed to combine the lost packets for different receivers. The expected number of retransmissions for a packet to reach every receiver successfully is calculated under different schemes. After comparison, it concludes that the scheme applying a dynamic network coding enjoys the smallest number of retransmissions. The following work in [88] focuses on finding the optimal coding set of the lost packets so that the number of retransmissions is minimised while ensuring that receivers can decode the lost packets upon receiving a coded packet. This coding set can be obtained by a colouring-based heuristic algorithm. The encoding of the lost packets relies on a packet-loss table, which contains the packet loss information of every receiver. The maintaining of the packet-loss table requires feedback from every receiver.

However, in wireless communication, especially in broadcast scenario, the feedback is expensive in terms of bandwidth and energy efficiency. Therefore, the retransmission schemes without feedback information are proposed in [89, 90, 91]. Additionally, the network coding schemes proposed in this thesis do not rely on feedback information.

In [89, 90], the authors consider a two stage broadcast scheme where every node broadcasts its native packet in the first stage while an XOR coded packet in the second stage. They investigate the optimal numbers of packets to be encoded in the second stage to minimise the expected number of retransmissions where the connection probabilities of every channel are given. However, the coding scheme in [89] does not always outperform non-coded network in terms of the expected number of retransmissions. In comparison, the schemes proposed in this thesis allows a node to choose not to perform coding when decoding cannot be performed because of lacking of native packets; therefore, they can achieve at least the same performance as the non-coded networks, which are better than that proposed in [89].

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Ref. [91] considers one-to-all model where one transmitter transmits multiple streams to multiple destinations over lossy channels. An XOR based retransmission scheme is proposed, aiming at providing fairness service to all users in terms of the service time and goodput. In the proposed scheme, the transmitter estimates the reception status of all receivers without ex-tra overheads. Then after comparing all possible coding sets, the scheduler selects the frames which offer the best performance under fairness constraints. Multiple selected frames are en-coded by XOR operation. This retransmission scheme is implemented in a real environment and its effectiveness is demonstrated.

Moreover, the reliability gain is characterised analytically in [92, 5], where network coding is compared with traditional error control protocols, such as ARQ and FEC. The considered systems have tree topologies where each multicast tree has equal number of children. The ex-pected numbers of retransmissions by the source node under different error control protocols are computed. Based on numerical comparison, it is conjectured that the reliability gain made by network coding increases logarithmically with respect to the number of receivers in a mul-ticast group compared with a simple ARQ scheme. This hypothesis is then proved in the latter work [5].

Combination of routing and network coding

Another approach to improve reliability is to combine routing with network coding [93, 94], where an individual node mixes different received packets heading towards the same destina-tion [95]. An example is given in Figure 2.5.

Suppose that node N1 needs to send three packets x, y and z to node N5. Figure 2.5 (a) illustrates the traditional method of sending these messages using three routes from node N1to node N5. After node N1sending three packets, node N3only receives x and y. Then, node N3 forwards both packets x and y to node N5, whereas node N5only receives y. Similarly, node N5 receives packets x and y from route N1-N2-N5and N1-N4-N5respectively. Finally, node N5has only packets x and y. Figure 2.5 (b) illustrates the case when both network coding and multi-path routing techniques are applied. Consider the same scenario as Figure 2.5 (a) in terms of packet loss, node N3only receives packets x and y from node N1. Then, node N3sends packet yand an encoded packet x ⊕ y to node N5, whereas node N5 only receives the second packet similarly to the case in Figure 2.5 (a). At last, node N5receives packets x, x ⊕ y and y ⊕ z, from

Design and Reliability Performance Evaluation of Network Coding Schemes for Lossy Wireless Networks

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Design and Reliability Performance Evaluation of

Network Coding Schemes for Lossy Wireless Networks

Li Ma

Abstract

Acknowledgements

Statement of originality

Related publications

Table of contents

List of figures

List of tables

List of symbols

List of abbreviations

Chapter 1

Introduction

1.1

Background

1.2

Research problems

1.3

Main contributions

1.4

Thesis outline

Chapter 2

Network Coding

2.1

Background

2.2

Network coding schemes

2.3

Benefits made by network coding